Differential 2-rigs

TL;DR

This paper introduces the concept of differential 2-rigs, categories with a differential structure that unify applications across computer science, topology, and combinatorics, revealing deep connections between these fields.

Contribution

It formalizes the notion of differential 2-rigs, constructs free examples, and establishes initiality results linking these categories to various mathematical and computational structures.

Findings

Defined differential 2-rigs with a Leibniz rule analogue

Constructed free differential 2-rigs from signatures

Proved initiality results connecting categories to applications

Abstract

We study the notion of a "differential 2-rig", a category R with coproducts and a monoidal structure distributing over them, also equipped with an endofunctor D : R -> R that satisfies a categorified analogue of the Leibniz rule. This is intended as a tool to unify various applications of such categories to computer science, algebraic topology, and enumerative combinatorics. The theory of differential 2-rigs has a geometric flavour but boils down to a specialization of the theory of tensorial strengths on endofunctors; this builds a surprising connection between apparently disconnected fields. We build "free 2-rigs" on a signature, and we prove various initiality results: for example, a certain category of colored species is the free differential 2-rig on a single generator.